Divide the following complex numbers: $\dfrac{5 e^{\pi i / 6}}{ e^{\pi i}}$ (The dividend is plotted in blue and the divisor in plotted in green. Your current answer will be plotted orange.)
Explanation: Dividing complex numbers in polar forms can be done by dividing the radii and subtracting the angles. The first number ( $5 e^{\pi i / 6}$ ) has angle $\frac{1}{6}\pi$ and radius 5. The second number ( $ e^{\pi i}$ ) has angle $1\pi$ and radius 1. The radius of the result will be $\frac{5}{1}$ , which is 5. The difference of the angles is $\frac{1}{6}\pi - 1\pi = -\frac{5}{6}\pi$ The angle $-\frac{5}{6}\pi$ is negative. A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $-\frac{5}{6}\pi + 2 \pi = \frac{7}{6}\pi$ The radius of the result is $5$ and the angle of the result is $\frac{7}{6}\pi$.